No can do.

Knowing, preparing, and casting are three different parts of the system, and have different rules binding them. It is especially important when multi-classing, because that's the only way you get access to more than one.

For the Wizard, Preparing and Casting Spells (PHB p114) explicitly refers to "wizard spells". The Your Spellbook sidebar also refers to "wizard spells".

In Multiclassing (PHB p164) it specifically addresses Spells Known and Prepared:

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

You may have a common pool of slots for casting, but there is no crossover between the two different types of spells for knowing or preparing. When putting spells into your replacement wizard spell book, you can only select from those you have prepared as wizard spells.

It's worth noting that you do not have to split your prepared spells between the two classes. Each of them determines prepared spells individually, based on the individual class level and the individual class' spellcasting attribute.